Ch 10: Interactions and Potential Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 10: Interactions and Potential Energy

Problem 24

Chapter 10, Problem 24

FIGURE EX10.24 is the potential-energy diagram for a 500 g particle that is released from rest at A. What are the particle's speeds at B, C, and D?

Verified step by step guidance

Step 1: Understand the problem. The potential-energy diagram shows the potential energy (U) of a 500 g particle as a function of position (x). The particle is released from rest at point A, and we need to find its speeds at points B, C, and D using the principle of conservation of energy.

Step 2: Apply the conservation of mechanical energy. The total mechanical energy (E) of the particle is the sum of its kinetic energy (K) and potential energy (U). Since the particle is released from rest at A, its initial kinetic energy is zero, and its total energy is equal to the potential energy at A: E = U_A.

Step 3: Determine the kinetic energy at each point. At any point (B, C, or D), the kinetic energy (K) can be found using the equation K = E - U, where U is the potential energy at that point. Use the values of U from the graph for points B, C, and D.

Step 4: Relate kinetic energy to speed. The kinetic energy of the particle is given by the formula:

K_{particle} = \frac{1}{2} m_{particle} v^{2}

. Rearrange this equation to solve for the speed (v):

v = \sqrt{\frac{2 K_{particle}}{m_{particle}}}

. Substitute the values of K and m to find the speed at each point.

Step 5: Perform the calculations for each point. Use the mass of the particle (m = 0.5 kg) and the potential energy values from the graph at points B, C, and D to calculate the kinetic energy and then the speed at each point. Ensure units are consistent throughout the calculations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Potential Energy

Potential energy (U) is the energy stored in an object due to its position in a force field, typically a gravitational or elastic field. In the context of the diagram, the potential energy varies with position, indicating how much energy the particle has at different points along its path. The higher the potential energy, the more work can be done by the particle when it moves to a lower potential energy position.

Conservation of Mechanical Energy

The principle of conservation of mechanical energy states that in the absence of non-conservative forces (like friction), the total mechanical energy of a system remains constant. This means that the sum of potential energy and kinetic energy at any point in the motion will equal the initial total energy. For the particle in the question, this principle allows us to relate its speeds at points B, C, and D to its potential energy at those points.

Kinetic Energy

Kinetic energy (K) is the energy of an object due to its motion, calculated using the formula K = 1/2 mv², where m is mass and v is velocity. As the particle moves through the potential energy landscape, its kinetic energy will change as it converts potential energy into kinetic energy and vice versa. By analyzing the potential energy at points B, C, and D, we can determine the corresponding speeds of the particle at those locations.

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